What is the speaker's claim?
The speaker's claim is that separation will not be end of the relationship he has with his love. The title of the poem, valediction, means a request of a command -- to forbid mourning. The poem was written to his wife because while she was pregnant, he needed to travel away from her, but the poem is a reassurance of his love for her and their love for each other.
He uses many many metaphors to suggest the singularity and specialness of their relationship, but the most unique comes in the last three stanzas. Previous to this he has said: don't make a scene when I leave; we are better than lovers who need "eyes, lips, and hands" to maintain their love; we are like gold that can be expanded to a great degree but doesn't break. In the end he compares the two of them to a compass -- the kind used to draw a perfect circle. He explains that they are always connected in the middle, and that she is the "fixed foot" in the center that allows him, the pencil / moving end, to move around and do what it needs to do, but that will "end where I begun" because she is the strong unmoved part. His claim is not only are they joined, but they work together in perfect harmony and one won't be right without the other.
The basis for the speaker’s claim in lines 13–20 is that the love of ordinary lovers (“sublunary,” line 13) is only physical and that for this reason they cannot retain their love at a distance. By contrast, the love of the speaker and the listener is not just a physical love, but is also an interaction of minds; therefore a separation of the physical does not imply a separation of the mental. This can be quite confusing to a modern reader, but this is quite common in John Donne's poetry: the use of metaphysical elements such as the mind versus the body or both the mind and body acting as one unit in the areas of love, wisdom, etc.